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Chapter 1: Problem 59

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -9 \quad-8 $$

Short Answer

Expert verified
-9 < -8

Step by step solution

01

Understanding the problem

We need to compare two numbers: -9 and -8, and determine the correct inequality sign (either \(<\) or \(>\)) that makes a true statement.
02

Analyzing Negative Numbers

When comparing negative numbers, remember that the larger number is the one that is less negative. This means it has a value closer to zero.
03

Compare -9 and -8

Since -9 is further from zero than -8, -9 is actually less than -8. So we state this as \(-9 < -8\).
04

Conclusion

The inequality \(-9 < -8\) correctly represents the relationship between the two numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Numbers
Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front of them. For instance, -9 is a negative number. It's important to understand that the more negative a number is, the further away it is from zero. This is in contrast to positive numbers which increase in value as they move away from zero.

A useful tip when thinking about negative numbers is to imagine them on a temperature scale. Consider -9 as being colder than -8, as it is lower down the thermometer. This perspective can help you remember that as negative numbers decrease, they become more negative.

When comparing negative numbers, keep in mind:
  • The number closer to zero is the larger number.
  • The further from zero, the smaller the number.
  • -8 is closer to zero than -9, so -8 is larger than -9.
Inequalities
Inequalities are mathematical expressions used to show how two numbers, expressions, or quantities relate to each other. The basic inequality symbols are:
  • \(<\) meaning "less than"
  • \(>\) meaning "greater than"
These symbols help indicate the relative size of numbers.

In the context of negative numbers, inequalities are a bit counterintuitive. With positive numbers, you might usually say 9 > 8 because 9 is larger. With negatives, however, you must remember that numbers decrease as their absolute value increases. Hence, for negative numbers, -9 < -8 because -9 is more negative than -8 and therefore smaller. Inequalities play a crucial role in ordering numbers and understanding number relationships.
Number Line Position
A number line is a straight, horizontal line with numbers placed at intervals along the length of the line. Numbers increase as you move right and decrease as you move left.

On a number line, you can clearly see the position of -9 and -8. -9 is to the left of -8, demonstrating that it is smaller than -8. The closer a number is to the left end of the line, the smaller it is. Conversely, numbers on the right side are larger.

Using a number line, you can
  • Visually compare two numbers.
  • Determine which number is larger or smaller quickly.
  • Understand the distance from zero, providing an idea of the number's size.
Using number lines is a simple and effective way to master inequalities, especially when dealing with negative numbers.

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